Recall that a cocomplete abelian category \(\mathcal {A}\)  is called a Grothendieck category if direct limits are exact in \(\mathcal {A}\)  and \(\mathcal {A}\)  has a generator. Let \(M\)  be an \(R\) -module. Denote by \(\sigma [M]\)  the the full subcategory
of \(\mathsf {Mod}\hspace{0.72229pt}R\)  whose objects are submodules of \(M\) -generated modules. Recall from [1]} that \(\sigma [M]\)  is a Grothendieck category with enough injective objects, see [1]}. For any family \(\lbrace T_{\lambda }\rbrace _{\lambda \in \Lambda }\)  of modules in \(\sigma [M]\) , the product in \(\sigma [M]\)  exists, see [1]}. If \(W\)  is an injective module in \(\mathsf {Mod}\hspace{0.72229pt}R\) , \(\mathsf {Tr}_{M}(W)\)  is an injective module in \(\sigma [M]\) . Moreover, if \(W\)  is a cogenerator for \(\mathsf {Mod}\hspace{0.72229pt}R\) , \(\mathsf {Tr}_{M}(W)\)  is a cogenerator for \(\sigma [M]\) . If the \(R\) -module \(M\)  is finitely generated as a module over \(\mathsf {End}_{R}(M)\) , then \(\sigma [M]=\mathsf {Mod}\hspace{0.72229pt}(R/\mathsf {ann}_{R}(M))\) , where \(\mathsf {ann}_{R}(M)=\lbrace r\in R~|~Mr=0\rbrace \) .
